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A069280 19-almost primes (generalization of semiprimes).

%I A069280 #32 Feb 16 2025 08:32:46
%S A069280 524288,786432,1179648,1310720,1769472,1835008,1966080,2654208,
%T A069280 2752512,2883584,2949120,3276800,3407872,3981312,4128768,4325376,
%U A069280 4423680,4456448,4587520,4915200,4980736,5111808,5971968,6029312,6193152
%N A069280 19-almost primes (generalization of semiprimes).
%C A069280 Product of 19 not necessarily distinct primes.
%C A069280 Divisible by exactly 19 prime powers (not including 1).
%H A069280 D. W. Wilson, <a href="/A069280/b069280.txt">Table of n, a(n) for n = 1..10000</a>
%H A069280 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F A069280 Product p_i^e_i with Sum e_i = 19.
%o A069280 (PARI) k=19; start=2^k; finish=8000000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v
%o A069280 (Python)
%o A069280 from math import prod, isqrt
%o A069280 from sympy import primerange, integer_nthroot, primepi
%o A069280 def A069280(n):
%o A069280     def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
%o A069280     def f(x): return int(n-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,19)))
%o A069280     kmin, kmax = 1,2
%o A069280     while f(kmax) >= kmax:
%o A069280         kmax <<= 1
%o A069280     while True:
%o A069280         kmid = kmax+kmin>>1
%o A069280         if f(kmid) < kmid:
%o A069280             kmax = kmid
%o A069280         else:
%o A069280             kmin = kmid
%o A069280         if kmax-kmin <= 1:
%o A069280             break
%o A069280     return kmax # _Chai Wah Wu_, Aug 23 2024
%Y A069280 Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), this sequence (r = 19), A069281 (r = 20). - _Jason Kimberley_, Oct 02 2011
%K A069280 nonn
%O A069280 1,1
%A A069280 _Rick L. Shepherd_, Mar 13 2002